Volume 33 Number 5 Summer 2001 Making
Teaching Public James D. Lehman, Janet Warfield, Michael Palm, and Terry
Wood Abstract Reform in mathematics education has been significantly influenced by the widespread notion that children construct mathematical understanding in social contexts, including the classroom. Empirical research has supported this position by providing evidence of the diversity in learning among children. Translating this theory of learning, however, and putting it into “real world” practice have proven difficult. The complexity of the pedagogy requires that teachers understand a variety of student learning strategies and help students progress in their thinking, both collectively and individually. This article reports on one portion of a two-year inservice professional development project focused on the mathematics teaching of eight elementary teachers. The project’s approach to professional development was based on prior theoretical and empirical research conducted in reform-based mathematics classes (Wood, 1997; Wood & Turner-Vorbeck, 1999, 2001). Its two main objectives were: finding a means of helping teachers develop their pedagogy in ways consistent with current reform recommendations and addressing the need to bring reform-based pedagogy to a wider inservice teacher population. One means of reaching this larger audience was through technology and the Internet. Our research was targeted specifically at exploring ways of using the Internet as a forum for professional discourse regarding mathematics teaching. The following questions guided this inquiry.
Approach to Professional Development It is widely accepted that mathematics teacher development should enable teachers to learn about the way children think (e.g., cognitively guided instruction; see Fennema et al., 1996) and incorporate this with situations for improving teachers’ mathematical understanding (e.g., Schifter, 2001; Warfield, 2001). Less understood are the ways that pedagogical research can be incorporated to help teachers learn about their teaching. Our approach to professional development is unique in that it combines knowledge about children’s mathematical thinking with knowledge about the complexities involved in teaching reform-based classes. It creates an environment in which educators are able to study their own classroom situations in ways that allow for private examination as well as public inquiry into their teaching practices. Technology plays an important role in our approach as a means of enabling educators to improve their learning and teaching skills through continued self-study. Videotape as a Tool for Examining Practice During the first year of the project, participants made video recordings of their lessons and used those recordings to examine events that occurred in their classrooms. Rather than use “preferred practice” tapes made by others, having teachers view their own sessions gave them an opportunity for introspection. We created a three-step procedure that teachers used to respond to their own tapes:
By keeping a written journal, it was anticipated that teachers would become skilled at examining student learning in ways that would influence their own thinking and teaching. For the first 12 months of the project, teachers videotaped their lessons and made entries in their journals approximately once a month; during the second year, they carried out these procedures approximately twice a month. Throughout the project, we as university-level mathematics educators watched the videotapes, read the journals, and commented in the journals. These comments were often in the form of questions. Teacher Working Sessions Videotape proved to be a useful tool for focusing teachers’ attention on classroom events and stimulating them to reflect on their own teaching practices. Group work sessions provided opportunities for public examination of videotapes and inquiry into both children’s thinking and teachers’ instructional practices. During the first year of the project, teachers attended a week-long session in the summer and then attended work sessions (varying in length from two to six hours) approximately once a month throughout the academic year. In the second year of the project, the number of sessions was decreased to two days in August (before the beginning of the school year) and four day-long sessions during the school year. We were interested in developing a sustained dialogue among the teachers about their teaching, a dialogue that could be maintained throughout the school year. Although the work sessions partially addressed this issue, time for teachers’ examination of videotapes in the sessions was limited. Our second objective, that of increasing our audience base, proved to be difficult as well. The work sessions stretched our staff resources and could not easily be adapted for large groups. Moreover, the centralized nature of the sessions also required that participants be located in relative close proximity to one another. In an attempt to overcome these limitations, we turned to the Internet in the project’s second year. Internet-Based Tools Many teacher educators view the Internet as a particularly promising vehicle for professional development, because it can support group discussion, accommodate teachers’ schedules, reduce teacher isolation, and increase teachers’ familiarity with technology (DiMauro & Jacobs, 1995; McMahon, 1997; Riel & Levin, 1990; Rud, 1995). Moreover, it has the potential to engage a broad cross-section of teachers from disparate locations. We viewed it as a possible tool for linking teachers in order to strengthen the general community of professional practice. To support the project’s activities during its second year, we created two Internet-based tools: a Web site and an electronic mail (e-mail) list. The Web site offered participants a reference point for project information, including a schedule of activities, mathematics problems used for online discussions, tips about using e-mail, and guidelines for completing journal entries. The Web site also contained an extensive set of links to Internet resources that teachers could use to gather ideas for mathematics teaching. Project participants were given an opportunity to post their photograph and biographical information on the site as well. The e-mail list was the key Internet-based tool in the project. The list, which was established and maintained on a computer at Purdue University, created a public forum that participants could use to communicate with the group. The list was available for asynchronous communication at any time and from any place that was convenient to the participants. Like most e-mail lists, messages were automatically distributed to all participants in the project. It also created a message archive so that we, or the participating teachers, could retrieve the messages on a monthly basis. Methodology A total of 12 individuals participated in the online community (Table 1): eight teachers and four project staff members (two mathematics educators, one technology educator, and one mathematics education graduate student).
The e-mail list was put into operation in mid-August 1997, before the start of the school year. Participants were introduced to the use of the list, e-mail, and the Internet and were provided with a handout detailing these Internet features. The technology educator was available to assist any participant who needed help getting started. The first messages were posted to the list near the end of August. Initially, access proved to be a barrier. At the beginning of the school year, only three of the eight participating teachers had direct access to a computer with an Internet connection. We helped arrange Internet access for the remaining five teachers through a local school service center that offered teachers low-cost Internet access. However, the process took longer than anticipated, and some participants experienced problems getting connected. The technology educator made house calls to two participants’ homes (one on multiple occasions) to resolve technical difficulties. Not until October were all of the participants able to get online and contribute to the discussion forum. As a result, there was relatively little activity and no formal structuring of the online dialogue at the outset. Once all of the participants were connected, the online discussion began. Participants established a goal of contributing at least twice a week. To account for differences in the grade levels and schools represented by the participants, it was decided to present common mathematics problems to all the classes and then adjust them according to student ability. These common problems, the students’ strategies for solving them, and the participant’s own pedagogy were discussed and analyzed through the e-mail forum. Participants commented on their experiences, engaged each other in discussion, and responded to inquiry and feedback from the mathematics educators. For their part, the mathematics educators focused on getting the participants to reflect on student learning strategies they observed in the classroom as well as their own teaching practices. Initially, this was accomplished through comments and questions about the online discussion and then later in direct response to what individual teachers had written in their journals. Data Collection and Analysis The analysis procedures were drawn from a qualitative-interpretative research paradigm. The phenomenon of the online forum was treated as a single case study. Messages to the online forum were automatically archived by the list host computer retrieved monthly. Throughout the study, the messages were printed, read, reread, and analyzed. Individual messages posted to the forum constituted the unit of analysis. Supporting data were gathered from discussions with participants about the online experience during the working sessions, personal communications with participants, and observations of the participants’ use of the technology. In more than nine months, eight participating teachers and four members of the project staff posted a total of 421 messages to the online forum. Basic descriptive information was tabulated for each message including: the message number (based on the order of posting to the list), the date of the message, its sender, links to other messages (i.e., replies), and the number of words in the message. Only original message content was considered for the word count and in the content analysis.1 Initial ideas for coding the content of the messages were drawn from a theoretical framework for teaching and learning in mathematics classrooms developed by Wood and Turner-Vorbeck (1999; 2001). This framework presents two dimensions of teaching and learning: one dimension describing the intellectual demand for students’ thinking and one dimension describing students’ increasing participation in class discussion. This framework posits that, as mathematics classrooms become more reform oriented, student thinking moves from reporting of problem solutions to inquiry into solutions and then to “argument” about solutions. According to the theoretical framework, participation moves from being teacher centered to student centered. From this, we hypothesized an increase in sophistication of teachers’ views of their children’s thinking about mathematics as the project progressed. Further, we hypothesized that the framework might also describe the teachers’ learning practices. We expected that the teachers’ approach might change from being one of simple reporting to inquiry and then to critique. In the process, we anticipated that there would be a movement from teacher-centered to more student-centered interactions in their classrooms. Codes were applied to the online forum messages. Through constant comparative analysis (Glaser & Strauss, 1967; Strauss & Corbin, 1990), categories were proposed, tested, and refined. In the end, eight coding categories emerged to classify the content of messages from the online forum. This coding scheme encompassed four basic dimensions: teachers’ reflections on their practice, teachers’ reflections on students’ thinking/understanding, non-teaching-related content, and use of the technology. Each dimension came to be represented by two coding categories. For example, “teachers’ reflections on their practice” was divided into “reporting of teaching practices” and “inquiry into teaching practice.” The eight final categories were not mutually exclusive; most messages contained content that fell into more than one category. The eight message content categories were:
Each individual message was classified using one or more of these categories. Overall statistics were generated (e.g., the frequency of different types of messages, average number of words per message), and messages were examined monthly for evidence of any broad patterns. Profiles were developed to describe the types of messages and the patterns of postings on a month-by-month basis. Profiles of individual participants were created to describe participant postings and to determine how the online conversation evolved over time. In those instances where there was evidence of participants’ inquiry into their own teaching practice and/or students’ understanding, interpretive analysis was used to explore the dynamics of the online conversation. Additional data from observations of and discussions with participants about the online technology were used to help interpret the use of the online environment in this project. Results and Discussion Overall List Usage As noted, a total of 421 messages were posted to the list during the period of this study. The average message contained approximately 111 words. Figure 1 illustrates the number of messages posted during each month of the project, divided into those submitted by the members of the project staff and those submitted by the participating teachers.2
Figure 1. Number of messages posted per month by staff and participating teachers. Two things are evident from Figure 1. First, there were clear peaks and valleys in the volume of conversation. Second, the staff, particularly the two mathematics educators (TW and JW), accounted for a significant proportion (nearly 40%) of the messages posted to the list. The up and down pattern of message postings is common for discussion lists and is certainly not surprising for a list involving teachers. Because of the delays in getting participants online, five of the eight teachers did not first contribute to the forum until October. As a result, activity in the beginning was relatively light. Fewer message postings also occurred around traditional break times: the winter break (December), spring break (March), and nearing the end of school (May). Participants’ Use of the Technology Participants tended to become more comfortable with the technology and more capable of using it during the year. Evidence of this can be seen in the overall pattern of comments about the online environment, the errors in its use, and the substance of comments about it. Figure 2 shows the pattern of technology messages and errors in the use of the technology by month.
Figure 2. Number of comments about the online environment and technology errors by month. As Figure 2 illustrates, comments about the online environment (TECH) peaked relatively early in the online discussion (in October and November) and fell off thereafter. A number of the early comments expressed frustration with the difficulties of using the technology. However, most participants eventually became relatively comfortable with the technology, and their comments reflected this. Shortly after the end of the November group meeting, BT posted a message to the forum saying, “I guess you could say I am addicted to this crazy thing because it is only 5:30 and I just left all of you, but I felt the need to talk.” Only a few minutes later, KA responded, “I’m slightly addicted myself. We’re able to access our mail! Yeah!”. Interestingly, the frequency of technology-use errors showed a somewhat delayed pattern relative to the comments about technology (Figure 2). The number rose fairly steadily to a peak in January (discounting the “slow” month of December) and then fell off to relatively low levels for the remainder of the study period. To some extent, the early increase in the number of errors paralleled the increase in the number of messages posted to the forum. As more messages were posted in October and November, two of the busiest months, there were more errors. Some of the increase in errors, paradoxically, occurred as a result of participants’ increasing proficiency. For example, when participants learned to embed a previous message in a reply, errors occurred because of the inability of the mailing list software to properly handle the embedded header information in the original message. Participants learned to cope with these problems over time. For example, instead of including all of a previous message in a reply, participants began to be selective about the content they included, leading to a decrease in errors caused by header information in the original message. Overall, the pattern of errors suggests that it took about five to six months for participants to reach a reasonable level of proficiency with e-mail. Content Analysis of Messages The high proportion of messages coming from the staff suggests that the mathematics educators did much to sustain the dialogue. Content analysis showed that many of the messages posted to the forum were procedural or informational in nature and that many of these messages came from the staff. In addition, the mathematics educators posted many messages intended to prompt the teachers to reflect on their student’s thinking or their own practices. It is questionable how well the online conversation would have been sustained without this direct and frequent involvement on their part. An overall breakdown of message content is shown in Figure 3. The most common category of message content was P/I. S/O comments were the second most common; this is not surprising given that participants had known one another as part of this project for a year prior to the launch of the online forum. The four categories of reflection (RTP, ITP, RSU, ISU) were relatively much less common, although together they accounted for a significant portion of the content. However, the mathematics educators’ questions and comments to participating teachers are included in these totals. Actual reflections by the teachers themselves were not common.
Figure 3. Breakdown of message content (all participants combined). These results are similar to those reported by McMahon (1997), who studied the Mathematics Learning Forums, three eight-week online courses involving 35 teachers in electronic discussions of students’ learning of mathematics and mathematics teaching. In that study, discussion messages tended to be brief, and overall activity in the online discussions was more than 40% below expectations of twice-weekly contributions. A majority of the dialogue (59%) was assignment related. Reflection of participants varied, but only 29% of participants posted at least one reflective message (McMahon). In our study, there was considerable variation among individual participants in the use of the online forum. Figure 4 shows the numbers of messages posted by each participant and the average number of words per message. In terms of the number of messages posted to the forum, the two mathematics educators posted the most messages. Among the participating teachers, SB and BT each posted a total of nearly 60 messages. These two participants were the only ones who contributed at a level approaching what we had envisioned at the beginning of the project. One participant, RS, never adjusted to using e-mail for communication and posted a total of only three messages to the forum. RS did not, therefore, participate in this aspect of the development approach. The remaining five teachers (WM, KA, WK, CJ, and IV) all posted between 22 and 29 messages, an average of fewer than one per week.
Figure 4. Number of messages and average number of word per message by participant. Message lengths varied considerably. Some participants, such as CJ and KA, tended to write short messages that were only half as long as those posted by BT and WK. The latter two participants had relatively high average numbers of words per message, in part because they submitted lengthy messages that described in detail the students’ problem-solving activities in their classrooms. Teachers’ Reflections The primary goals of the online forum were to promote thoughtful discourse and to increase community among the teachers. In the analysis of the messages, two main discussion threads were identified: exchanges regarding students’ thinking/understanding and exchanges about teaching practice. Some messages contained elements of both discussions while others were limited to just one or the other. Each of the discussion threads was subdivided and coded as either whether it showed evidence of inquiry (questioning, clarifying, explaining) or not. Figure 5 shows the overall pattern of message postings related to student thinking by month, and Figure 6 presents the pattern for messages related to discussions of teaching practice.
Figure 5. Messages that included discussions related to student thinking by month.
Figure 6. Messages that included discussions related to teaching practice by month. The overall patterns of reflective dialogue reveal some information about the forum. As noted previously, each category of reflection was observed less frequently than routine procedural and information messages (P/I) and social or other messages (S/O). Figures 5 and 6 show that it was more common to see descriptions or mere reporting of classroom events than it was to see evidence of inquiry. In other words, teachers simply described classroom events more often than they analyzed those events. The data show that between the two types of discussions, there were some differences in overall message patterns. There was greater consistency from month to month in the numbers of messages regarding discussions about student thinking. This is likely because the online discussion was intentionally focused on student thinking and revolved around consideration of common mathematics problems given in the participants’ classrooms. The discussion moved from there, generally with some prodding from the mathematics educators, to a consideration of pedagogical issues. Frequency of messages about teaching practice showed a bimodal pattern, with peaks occurring early in the year and again late in the year. The early peak corresponded with the participants’ increasing level of comfort with using the forum as a means of communication. As they became more comfortable, participants seemed to enjoy the chance to interact with one another about teaching issues. However, as the novelty wore off, reflective dialogue waned. The late peak corresponded to an assertive effort in April by the mathematics educators to challenge the teachers to address pedagogical issues. The regularity of postings concerning student thinking indicates that the online forum was successful in fostering this type of dialogue. However, it was more difficult for the teachers to sustain a dialogue about their own practices. Inquiry into Student Thinking The importance of teachers attending to detail in their students’ mathematical thinking has been well documented (cf., Fennema et al., 1996; Schifter, 2001; Warfield, 2001). From the beginning of the project, we asked the participants to describe classroom discourse in their journal entries and discuss their classroom videotapes in the working sessions. This approach was effective, in part, because the mathematics educators who read the journals and the other teachers who discussed the students’ thinking in the working sessions had access to the videotapes. It proved more difficult for teachers to use e-mail to describe students’ thinking in enough detail so that it could be understood by others without access to the classroom or the videotape. It was common for the mathematics educators to prompt the participants to go into depth about their students’ thinking. For example, the first common problem that participants explored in their classes was: “There are some cows and chickens in the field. Altogether there are __ heads and __ legs. How many of the animals are cows and how many are chickens?”3 One of the participants, WK, posted the following description of events in her classroom when students tried to solve this problem. The first pair to discuss their solution wrote on the OHP [overhead projector]:
This description prompted one of the mathematics educators, JW, to intervene to get additional information about what had transpired.4 She wrote:
Subsequently, WK replied to this prompting and provided additional information about what had transpired in her class.
This example illustrates intervention by a mathematics educator to get a participant to say more about the mathematics understanding that students were constructing in the classroom, and it illustrates how the forum was used for this sort of exploration. Inquiry into Teaching Practice Before the introduction of the online forum, the mathematics educators watched the teachers’ classroom videotapes, read their journals, and made comments in the journals regarding both the videos and what the teachers had written. Only the individual teacher to whom the comments were directed ever saw them. There was no regular procedure put in place to enable the teachers to respond. In March of the second year of the project, we asked teachers about the possibility of responding to their videos and journals through e-mail. We also asked if those responses should be sent only to individuals or to the whole list. The teachers were amenable to having the responses come through e-mail and preferred that they be sent to the whole group. This new method of communication gave teachers yet another opportunity for introspective thinking. Participant BT was the teacher who most deeply examined her own teaching practices as a result of the discussion on the online forum. One exchange began with a question from TW and JW in response to one of BT’s videotapes and her journal entry regarding that tape. The mathematics educators had been concerned that BT consistently interfered with students’ explanations of their thinking by stepping in and taking over the explanation. An episode on videotape along with a comment in her journal provided them an opportunity to raise the issue.
This set of messages, spanning more than a month of online dialogue, makes it clear that BT was able to confront some of her basic beliefs about teaching and the role of the teacher. She acknowledged that students must have responsibility for their learning, which seemed to be a big step toward a more reform-oriented pedagogy. These sorts of discussion are common when mathematics educators are present in teachers’ classrooms on an ongoing basis. Here, the use of the online forum created the opportunity for this ongoing discussion of a particular pedagogical issue. Summary and Conclusions This project sought to investigate the use of the Internet, primarily an online discussion forum, as a vehicle for supporting public inquiry into teaching. Some of the findings were disappointing. The teachers’ levels of participation in the project fell below expectations, and most of their messages contained procedural/informational or social content rather than any meaningful discussion of student thinking. Consistent facilitation by the mathematics educators was necessary to continue the dialogue and keep it focused. Rarely did it take on a spontaneous or self-perpetuating character. Factors that may have inhibited the discussion include:
Research indicates that a number of factors affect the success of online communities. McMahon (1997), who studied mathematics forums similar to ours, found, as did we, that facilitation promoted participation. She also found that certain participant characteristics were associated with greater online participation including time to use the online medium (i.e., reduced workload) and access to a computer at home. Riel and Levin (1990) reported that online communities tended to be more successful when:
Several factors weighed against the success of this online community. Although it was built around an existing group with a specific goal to explore mathematics teaching, the need for electronic communication was not compelling. Participants were not required by circumstances to interact electronically, and they were used to interacting with one another in person, having had a year of group meetings prior to the introduction of the e-mail list. For some, electronic participation was just another thing that had to be done and, hence, was often avoided. Regular patterns of participation did not develop. Only two of the eight teachers came close to meeting our goal of participating twice per week. Although access to the technology was easy for most of the participants, it was not universally convenient. Two of the least active participants, CJ and IV, had some access difficulties; CJ used a computer at school (accessible only outside normal schools hours) rather than one at home, and IV used her brother’s computer in his home. This lack of convenient access might have directly hampered their participation, or it may have been a convenient excuse not to participate more often. Because the success of online communities requires a sufficient “critical mass” of participants to sustain dialogue, the small number of participants in our study may have been inhibitory. Although they did not specifically identify group size as a factor, when Riel and Levin (1990) compared online communities, smaller online communities were always the ones judged to be least successful. Finally, the nature of online discussion itself probably played a role in the lack of success. Although the teachers embraced the use of the online environment for sharing ideas or tips about teaching, they were less inclined to critique themselves publicly or examine others. Other findings were positive. Within five to six months, most participating teachers became comfortable with and capable of using the Internet as a vehicle for communication. The use of common mathematics problems, given across different classes, provided an effective way to stimulate discussions among teachers representing different grade levels and schools. Mathematics educators were able to promote dialogue by asking focused questions and making relevant comments. In some instances, this intervention motivated participants to reconsider their beliefs and teaching practices. These enlightening exchanges were particularly successful when the mathematics educators used their knowledge of a teacher’s practice (gained through a variety of media including videotapes, journal entries, discussions in working sessions, and previous e-mail conversations) as a means of beginning and sustaining the dialogue with that teacher. We speculate that, had these conversations been carried out in a public forum over a longer period of time, the teachers might have become more able to ask the same kinds of questions themselves. Can the Internet be used successfully for teacher development and the creation of a community of reflective practice? Although our study does not provide a clear answer, the characteristics of the medium and the growing success of online courses suggest that the Internet can and should be an excellent tool for teacher development and community building. But, the practical realities are challenging. More research is needed to better understand the factors that can lead to the successful development of online communities of practice. Acknowledgement This research was supported by the National Science Foundation under award RED 925-4939. All opinions expressed are those of the authors. Contributors James D. Lehman is a professor of educational technology at Purdue University. Janet Warfield is an assistant professor of mathematics education at Purdue University. Michael Palm recently completed his master’s degree in mathematics education from Purdue University and now teaches in Hawaii. Terry Wood is a professor of mathematics education at Purdue University. Contact James D. Lehman References DiMauro, V., & Jacobs, G. (1995). Collaborative electronic network building. Journal of Computers in Mathematics and Science Teaching, 14(1/2), 119–131. Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, B., & Empson, S. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27, 403–434. Glaser, B., & Strauss, A. (1967). The discovery of grounded theory. Chicago: Aldine. McMahon, T. A. (1997, March). From isolation to interaction? Network-based professional development and teacher professional communication. Paper presented at the annual meeting of the American Educational Research Association, Chicago. Riel, M. M., & Levin, J. A. (1990). Building electronic communities: Success and failure in computer networking. Instructional Science, 19, 145–169. Rud, A. G. (1995, April). The development of an online community of inquiry. Paper presented at the annual meeting of the American Educational Research Association, San Francisco. Schifter, D. (2001). Learning to see the invisible. In T. Wood, B. S. Nelson, & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 109–134). Mahwah, NJ: Erlbaum Associates. Strauss, A., & Corbin, J. (1990). Basics of qualitative research. Newbury Park, CA: Sage Publications. Warfield, J. (2001). Where mathematics content knowledge matters: Learning about and building on children’s mathematical thinking. In T. Wood, B. S. Nelson, & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 135–156). Mahwah, NJ: Erlbaum Associates. Wood, T. (1997, July). Creating classroom interactions for mathematical reasoning: Beyond “natural” teaching. Paper presented at the 49th conference of the International Commission for the Study and Improvement of Mathematics Education (CIEAEM), Setubal, Portugal. Wood, T., & Turner-Vorbeck, T. (1999). Developing teaching of mathematics: Making connections in practice. In L. Burton (Ed.), Learning mathematics: From hierarchies to networks (pp. 173–186). London: Falmer Press. Wood, T., & Turner-Vorbeck, T. (2001). Extending the conception of mathematics teaching. In T. Wood, B. S. Nelson, & J. Warfield (Eds.), Beyond classical pedagogy: Teaching elementary school mathematics (pp. 185–208). Mahwah, NJ: Erlbaum Associates. Endnotes
A PDF file of the full article is available. Contact: jrte@iste.org. Please specifiy Volume and Issue number and article name. Copyright © 2001, ISTE (International Society for Technology in Education). All rights reserved. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
